SOLVING PROPORTIONS

This Lesson follows from Lesson 17. Here, we will see how to solve any proportion.

What is taught in most textbooks these days as ratio and proportion, is not. It is algebra. The student is taught to represent a ratio as a fraction, write the letter x for the unknown term, cross-multiply and solve an equation. That is an algebraic calculator.

A ratio requires understanding. It has meanng and is not merely symbolic. Because of that, the topic of ratio and proportion is educational.

"If we multiply two numbers by the same number, then the products will have the same ratio as the numbers we multiplied."

(Euclid, VII. 17.)

We have already seen that a ratio will be preserved if we divide both terms by the same number.

Example 5. Complete this proportion:

6 : 7 = ? : 28

Solution. 7 has been multiplied by 4 to give 28. Therefore, 6 also must be multiplied by 4:

6 : 7 = 24 : 28.

To solve that proportion --

6 : 7 = ? : 28

-- we could say:

"7 goes into 28 four times. Four times 6 is 24."

All the Examples and Problems in this lesson should be simple, mental calculations.

Example 6. Solve this proportion:

2 : 3 = 12 : ?

Solution. "2 goes into 12 six times. Six times 3 is 18."

2 : 3 = 12 : 18.

In fact, consider these columns of the multiples of 2 and 3:

23

46

69

812

1015

1218

1421

And so on.

Now, 2 is two thirds of 3. (Lesson 17.) And each multiple of 2 is two thirds of that same multiple of 3:

4 is two thirds of 6.

6 is two thirds of 9.

8 is two thirds of 12.

And so on. In fact, those are the only natural numbers where the first will be two thirds of the second.

Note that each pair have a common divisor. And upon dividing by that divisor, the quotients in every case are 2 and 3. That is the theorem of the common divisor. 2 and 3 are the lowest terms. They are the smallest numbers which have the ratio "two thirds."

Example 7. Name three pairs of numbers such that the first is three fifths of the second.

Solution. The elementary pair are 3 and 5. To generate others, take the same multiple of both: 6 and 10, 9 and 15, 12 and 20, and so on.

Example 8. 27 is three fourths of what number?

Solution. Proportionally:

3 : 4 = 27 : ?

"3 goes into 27 nine times. Nine times 4 is 36."

3 : 4 = 27 : 36.

27 is three fourths of 36.

Only a multiple of 3 can be three fourths of another number, which must be that same multiple of 4.